Question

Analysis of the complex numbers,
http://screencast.com/t/kZ8lzovtXsy6

Answer 1

w=x+iy
|w| = 1
so, x^2+y^2 =1
right ?

Answer 2

can you find 1/w ?

Answer 3

1/(x+iy) =... ?

Answer 4

1/(x+iy) = 1/w

Answer 5

yeah
multiply and divide by the conjugate of x+iy

Answer 6

oh

Answer 7

1/(x-iy)

Answer 8

or not

Answer 9

nopes

Answer 10

x - iy/x^2 - (iy)^2

Answer 11

1/(x+iy) = (x-iy)/ (x+iy)(x-iy)
what does denominator simplify to ?

Answer 12

yeah go on

Answer 13

w/x^2 - (iy)^2

Answer 14

nah

Answer 15

the numerator is NOT w

Answer 16

-w ?

Answer 17

what about the denominator ?

Answer 18

nopes, not -w
concentrate on denominator

Answer 19

ok..

Answer 20

so we have x^2 - (iy)^2

Answer 21

yes
(iy)^2 = i^2y^2 = ... ?

Answer 22

-y

Answer 23

x*w

Answer 24

denominator ?

Answer 25

ops -y^2

Answer 26

yes

Answer 27

so, x^2 - (-y^2) = ...

Answer 28

x^2 + y^2

Answer 29

=..?

Answer 30

obs(w) ?

Answer 31

see my very first comment

Answer 32

ironically obs(w) = 1 :D

Answer 33

yes
so denominator = 1

Answer 34

so, 1/w = x-iy

Answer 35

w+1/w =... ?

Answer 36

x-iy + x^2 + y^2/ x - iy

Answer 37

wot ?

Answer 38

w = x+iy
1/w = x-iy
isn't it ?

Answer 39

wait I am kind of infused as to why the sing changes with the fraction

Answer 40

confused*

Answer 41

we just proved it
and w and 1/w are conjugates of each other only when abs(w) = 1

Answer 42

that is when x^2+y^2 = 1

Answer 43

else, 1/w is just (x-iy)/(x^2+y^2)

Answer 44

exactly its (x-iy)/(x^2+y^2)

Answer 45

yes
and x^2+y^2 = 1 is given in this case

Answer 46

(x-iy)=1 <---- where is this given ?

Answer 47

we never said x-iy =1

Answer 48

ohhh I see

Answer 49

did i ever say that ?

Answer 50

ok ok I follow you now

Answer 51

nah I messed it

Answer 52

cool
w = x+iy
1/w = x-iy
isn't it ?

Answer 53

yes

Answer 54

w+1/w =

Answer 55

x+iy + 1/x+iy

Answer 56

I am sorry if I sound dummy I am completely new to complex analysis :D

Answer 57

still
1/w we proved it to be x-iy
right ?

Answer 58

or x+iy + x - iy /x+iy

Answer 59

(x+iy + 1)*(x - iy)